Device-independent tests of quantum channels

Abstract

We develop a device-independent framework for testing quantum channels. That is, we falsify a hypothesis about a quantum channel based only on an observed set of input–output correlations. Formally, the problem consists of characterizing the set of input–output correlations compatible with any arbitrary given quantum channel. For binary (i.e. two input symbols, two output symbols) correlations, we show that extremal correlations are always achieved by orthogonal encodings and measurements, irrespective of whether or not the channel preserves commutativity. We further provide a full, closed-form characterization of the sets of binary correlations in the case of: (i) any dihedrally covariant qubit channel (such as any Pauli and amplitude-damping channels) and (ii) any universally-covariant commutativity-preserving channel in an arbitrary dimension (such as any erasure, depolarizing, universal cloning and universal transposition channels).

1. Introduction

Any physical experiment is based upon the observation of correlations among events at various points in space and time, along with some assumptions about the underlying physics. Naturally, in order to be operational any such assumption must have been tested as a hypothesis in a previous experiment. Ultimately, to break an otherwise circular argument, experiments involving no further assumptions are required—that is, device-independent tests.

Formally, a hypothesis consists of a circuit,¹ which is usually assumed to have a global causal structure (following special relativity), and its components, which are usually assumed to be governed by classical or quantum theories and thus representable by channels.

Denoting a hypothesis (circuit) by $\mathcal{X}$, the set of correlations compatible with $\mathcal{X}$ is denoted by $S(\mathcal{X})$. Then, hypothesis $\mathcal{X}$ is falsified, along with any other hypothesis $\mathcal{Y}$ such that $S(\mathcal{Y})\subseteq S(\mathcal{X})$, as soon as the observed correlation does not belong to $S(\mathcal{X})$ (This inclusion relation induces an ordering among channels which is reminiscent of that introduced by Shannon [5] among classical channels). Therefore, from the theoretical viewpoint, the problem of falsifying a hypothesis $\mathcal{X}$ can be recast² as that of characterizing the set $S(\mathcal{X})$ of compatible correlations.

Since (discrete, memoryless) classical channels are by definition input–output correlations (conditional probabilities), the characterization of $S(\mathcal{X})$ is trivial in classical theory as it is a polytope easily related to the correlation defining the channel. By contrast, the problem is far from trivial in quantum theory: due to the existence of superpositions of states and effects, the set $S(\mathcal{X})$ can be strictly convex.

In this work, we address the problem of device-independent tests of quantum channels, in particular, the characterization of the set $S_m^n(\mathcal{X})$ of m-inputs/n-outputs correlations p_j|i obtainable through an arbitrary given channel $\mathcal{X}$, upon the input of an arbitrary preparation ${\{{\rho }_i\}}_{i=0}^{m-1}$ and the measurement of an arbitrary positive-operator valued measure (POVM) ${\{{\pi }_j\}}_{j=0}^{n-1}$, that is

1.1

The analogous problems of device-independent tests of quantum states and measurements have been recently addressed in [12,13], respectively.

An alternative formulation for the problem considered here can be given in terms of a ‘game’ involving two parties: an experimenter, claiming to be able to prepare quantum states, feed them through some quantum channel $\mathcal{X}$, and then perform measurements on the output, and a skeptical theoretician, willing to trust observed correlations only. If the experimenter produces some correlations lying outsides of $S_m^n(\mathcal{X})$, then the theoretician must conclude that the actual channel ${\mathcal{X}}^{\prime }$ is not worse than $\mathcal{X}$ at producing correlations, but this is not sufficient to support the experimenter’s claim. Indeed, in order to convince the theoretician, the experimenter must produce the entire set $S_m^n(\mathcal{X})$: in fact, it is sufficient to produce a set of correlations whose convex hull contains $S_m^n(\mathcal{X})$. Then, the theoretician must conclude that whatever channel the experimenter actually has is at least as good as $\mathcal{X}$ at producing correlations, and the experimenter’s claim is accepted.

It is hence clear that the problem of device-independent tests of quantum channels induces a preordering relation among quantum channels: $\mathcal{X}⪰\mathcal{Y}$ if and only if $S_m^n(\mathcal{X})\supseteq S_m^n(\mathcal{Y})$. (The order also depends upon m and n, but for compactness we drop the indexes whenever they are clear from the context). In order to characterize such preorder, for any given channel $\mathcal{X}$, we need to (i) provide the experimenter with all the states and measurements generating the extremal correlations of $S_m^n(\mathcal{X})$ and (ii) provide the theoretician with a full closed-form characterization of the set $S_m^n(\mathcal{X})$ of compatible correlations.

As a preliminary result, we find that the sets $S_m^n(\mathcal{X})$ coincide for any d-dimensional unitary and dephasing channels, for any d, m and n (this is an immediate consequence of a remarkable result by Frenkel & Weiner [14]). Upon considering only the binary case m=n=2, our first result is to show that any correlation on the boundary of $S_2^2(\mathcal{X})$ is achieved by a pair of commuting pure states—irrespective of whether $\mathcal{X}$ is a commutativity-preserving channel. Then, we derive the complete closed-form characterization of $S_2^2(\mathcal{X})$ for: (i) any given dihedrally covariant qubit channel, including any Pauli and amplitude-damping channels; and (ii) any given universally covariant commutativity-preserving channel, including any erasure, depolarizing, universal 1→2 cloning [15], and universal transposition [16] channels.

Upon specifying $\mathcal{X}$ as the d-dimensional identity channel ${\mathcal{I}}_d$, one recovers device-independent dimension tests analogous to those discussed in [17–20], in which case the aforementioned ordering induced by the inclusion $S_m^n({\mathcal{I}}_{d_0})\subseteq S_m^n({\mathcal{I}}_{d_1})\iff d_0\le d_1$ is of course total. However, the completeness of our characterization of $S_2^2(\mathcal{X})$ implies that our framework detects all correlations incompatible with the given hypothesis, unlike the studies [17–21] where the set of correlations is tested only along an arbitrarily chosen direction.

Let us provide a preview of some consequences of our results:

• — any Pauli channel ${\mathcal{P}}^{\mathit{\lambda }}:\rho \to {\mathrm{\lambda }}_0\rho +\sum _{k=1}^3{\mathrm{\lambda }}_k{\sigma }_k\rho {\sigma }_k^{†}$ is compatible with p if and only if

  $$\frac{|p_{1|1}-p_{1|2}|}{1-|p_{1|1}-p_{2|2}|}\le \underset{k\in [1,3]}{max}|2({\mathrm{\lambda }}_0+{\mathrm{\lambda }}_k)-1|;$$
• — any amplitude-damping channel ${\mathcal{A}}^{\mathrm{\lambda }}:\rho \to A_0\rho A_0^{†}+A_1\rho A_1^{†}$ with $A_0=|0⟩⟨0|+\sqrt{\mathrm{\lambda }}|1⟩⟨1|$ and $A_1=\sqrt{1-\mathrm{\lambda }}|0⟩⟨1|$ is compatible with p if and only if

  $${(\sqrt{p_{1|2}{p}_{2|1}}-\sqrt{p_{1|1}{p}_{2|2}})}^2\le \mathrm{\lambda };$$
• — any d-dimensional erasure channel ${\mathcal{E}}_d:\rho \to \mathrm{\lambda }\rho \oplus (1-\mathrm{\lambda })\mathrm{Tr}[\rho ]\varphi$ for some pure state ϕ is compatible with p if and only if

  $$|p_{1|1}-p_{1|2}|\le \mathrm{\lambda };$$
• — any d-dimensional depolarizing channel ${\mathcal{D}}_d^{\mathrm{\lambda }}:\rho \to \mathrm{\lambda }\rho +(1-\mathrm{\lambda })\mathrm{Tr}[\rho ]𝟙/d$ is compatible with p if and only if

  $$|p_{1|1}-p_{1|2}|\le \mathrm{\lambda }$$

  and

  $$\frac{|p_{1|1}-p_{1|2}|}{1-|p_{1|1}-p_{2|2}|}\le \frac{d\mathrm{\lambda }}{2-2\mathrm{\lambda }+d\mathrm{\lambda }};$$
• — the d-dimensional universal optimal 1→2 cloning [15] channel ${\mathcal{C}}_d$ is compatible with p if and only if

  $$|p_{1|1}-p_{1|2}|\le \frac{d}{d+1};$$
• — any d-dimensional universal optimal transposition [16] channel ${\mathcal{T}}_d$ is compatible with p if and only if

  $$|p_{1|1}-p_{1|2}|\le \frac{1}{d+1}$$

  and

  $$\frac{|p_{1|1}-p_{1|2}|}{1-|p_{1|1}-p_{2|2}|}\le \frac{1}{3}.$$

This paper is structured as follows. We will introduce our framework and discuss the case of unitary and trace class channels in §2. For the binary case, introduced in §3, we will solve the problem for any qubit dihedrally-covariant channel in §4, and for any arbitrary-dimensional universally covariant commutativity-preserving channel in §5. In §6, we will provide a natural geometrical interpretation of our results and in §7, we will summarize our results and present further outlooks.

2. General results

We will make use of standard definitions and results in quantum information theory [22]. Since $S_m^n(\mathcal{X})$ is convex for any n and m, the hyperplane separation theorem [23,24] states that $p\notin S_m^n(\mathcal{X})$ if and only if there exists an m×n real matrix w such that

$$p^{\mathrm{T}}\cdot w-W(\mathcal{X},w)>0,$$ 2.1

where $p^{\mathrm{T}}\cdot w:=\sum _{i,j}{p}_{j|i}{w}_{i,j}$ and

$$W(\mathcal{X},w):=\underset{q\in S_m^n(\mathcal{X})}{max}{w}^{\mathrm{T}}\cdot q.$$ 2.2

We call w a channel witness and $W(\mathcal{X},w)$ its threshold value for channel $\mathcal{X}$.

Although equation (2.1) generally allows one to detect some conditional probability distributions p not belonging to $S_m^n(\mathcal{X})$ for any arbitrarily fixed witness w, here our aim is to detect any such p. Direct application of equation (2.1) is impractical, as one would need to consider all of the infinitely many witnesses w. Note, however, that equation (2.1) can be rewritten through negation by stating that $p\in S_m^n(\mathcal{X})$ if and only if for any m×n witness w one has

$$p^{\mathrm{T}}\cdot w-W(\mathcal{X},w)\le 0.$$

We then have our first preliminary result.

Lemma 2.1 —

A channel $\mathcal{X}:\mathcal{L}(\mathcal{H})\to \mathcal{L}(\mathcal{K})$ is compatible with conditional probability distribution p if and only if

$$\underset{w}{max}[p^{\mathrm{T}}\cdot w-W(\mathcal{X},w)]\le 0.$$ 2.3

Let us start by considering an arbitrary d-dimensional unitary channel ${\mathcal{U}}_d:\rho \to U\rho U^{†}$, for some unitary $U\in \mathcal{L}(\mathcal{H})$ with $\dim \hspace{0.17em}\mathcal{H}=d$. If d≥m, the maximization in equation (2.2) is trivial, since the input labels i∈[1,m] can all be encoded on orthogonal states, so that any m×n conditional probability distribution q can in fact be obtained. However, if d<m, the evaluation of the witness threshold $W({\mathcal{U}}_d,w)$ for any witness w is far from obvious. The solution immediately follows from a recent, remarkable result by Frenkel & Weiner [14]. It turns out that $W({\mathcal{U}}_d,w)$ is attained on extremal conditional probability distributions q compatible with the exchange of a classical d-level system, namely, those q where q_j|i=0 or 1 for any i and j, and such that q_j|i≠0 for at most d different values of j. Frenkel and Weiner’s result hence guarantees that the threshold $W({\mathcal{U}}_d,w)$ can be provided in closed form since, for any m and n, the number of such extremal classical conditional probabilities is finite, i.e. the set $S_m^n({\mathcal{U}}_d)$ is a polytope. Any probability p lying outside $S_m^n({\mathcal{U}}_d)$ can thus be detected by testing the violation of equation (2.3) for a finite number of witnesses w, corresponding to the faces of the polytope. Moreover, the set $S_m^n({\mathcal{U}}_d)$ of distributions compatible with any d-dimensional unitary channel ${\mathcal{U}}_d$ coincides with the set $S_m^n({\mathcal{F}}_d^{\mathrm{\lambda }})\phantom{{\hspace{0.17em}}_{j_j}}$ of distributions compatible with any d-dimensional dephasing channel ${\mathcal{F}}_d^{\mathrm{\lambda }}:\rho \to \mathrm{\lambda }\rho +(1-\mathrm{\lambda })\sum _k⟨k|\rho |k⟩|k⟩⟨k|$.

At the opposite end of the unitary channels, there sit trace-class channels $\mathcal{T}:\rho \to \sigma$ for some arbitrary but fixed state σ. In this case, no information about i (the input label) can be communicated. Of course, the set $S_m^n(\mathcal{T})$ of correlations achievable through any trace-class channel $\mathcal{T}$ does not depend on the particular choice of σ: a trace-class channel simply means that no communication is available. For any trace-class channel $\mathcal{T}$ and any witness w, it immediately follows that the threshold $W(\mathcal{T},w)$ is achieved by conditional probabilities q such that q_j|i=1 for a single value of j, and therefore is given by $W(\mathcal{T},w)=\underset{j}{max}\sum _i{w}_{i,j}$. As a consequence, the set $S_m^n(\mathcal{T})$ is a polytope with n vertices, and any probability p lying outside $S_m^n(\mathcal{T})$ can be detected by testing the violation of equation (2.3) for a finite number of witnesses w.

3. Binary conditional probability distribution

In the remainder of this work, we will consider the case where p is a binary input–output conditional probability distribution (i.e. m=n=2).

First, we show that it suffices to consider diagonal or anti-diagonal witnesses with positive entries summing up to one. Indeed, for any witness w, the witness w′:=α(w+β), where α>0 and β is such that β_i,j is independent of j, leaves equation (2.3) invariant for any conditional probability distribution p and channel $\mathcal{X}$, since $w^{\prime }\cdot p=\alpha (p^{\mathrm{T}}\cdot w+\sum _i{\beta }_{i,1})$.

By taking ${\beta }_{i,j}=-\underset{k}{min}{w}_{i,k}$ for any i and j, the witness w′ is diagonal, anti-diagonal, or has a single non-null column. We first consider the latter case. Clearly, the maximum in equation (2.2) is attained when p is a vertex of the polytope $S_2^2(\mathcal{T})$ of probabilities compatible with any trace-type channel $\mathcal{T}$, and therefore equation (2.3) is always verified. Then we consider the case of diagonal and anti-diagonal witnesses. By taking ${\alpha }^{-1}=\sum _i|w_{i,1}-w_{i,2}|$ one recovers the normalization condition $\sum _{i,j}{w}_{i,j}=1$, thus proving the statement.

Therefore, upon denoting with w^±(ω) the diagonal and anti-diagonal witnesses given by

$$w^{+}(\omega ):=\left(\begin{array}{cc}\frac{1+\omega }{2}& 0\\ 0& \frac{1-\omega }{2}\end{array}\right)\mathrm{and}{w}^{-}(\omega ):=\left(\begin{array}{cc}0& \frac{1+\omega }{2}\\ \frac{1-\omega }{2}& 0\end{array}\right),$$

where ω∈[−1,1], one has the following preliminary result.

Lemma 3.1 —

The maximum in equation (2.3) is attained for a diagonal or anti-diagonal witness, namely

$$\underset{w}{max}(p^{\mathrm{T}}\cdot w-W(\mathcal{X},w))=\underset{\omega \in [-1,1]}{max}(p^{\mathrm{T}}\cdot w^{\pm }(\omega )-W(\mathcal{X},w^{\pm }(\omega ))).$$

Any extremal distribution q in equation (2.2) can be represented by states ρ₀ and ρ₁ and a POVM {π₀,π₁} such that $q_{j|i}=\mathrm{Tr}[\mathcal{X}({\rho }_i){\pi }_j]$. Since w^±(ω) is diagonal or anti-diagonal, equation (2.2) represents the maximum probability of success in the discrimination of states {ρ₀,ρ₁} with prior probabilities given by the non-null entries of w, in the presence of noise $\mathcal{X}$, namely

$$W(\mathcal{X},w^{\pm }(\omega ))=\frac{1}{2}\underset{\begin{array}{c}{\rho }_0,{\rho }_1\\ \{{\pi }_0,{\pi }_1\}\end{array}}{max}[(1+\omega )\mathrm{Tr}[\mathcal{X}({\rho }_0){\pi }_0]+(1-\omega )\mathrm{Tr}[\mathcal{X}({\rho }_1){\pi }_1]].$$

It is a well-known fact [25] that the solution of the optimization problem over POVMs is given as a function of the Helstrom matrix defined as

$$H_{\omega }({\rho }_0,{\rho }_1):=\frac{1+\omega }{2}{\rho }_0-\frac{1-\omega }{2}{\rho }_1,$$

as follows

$$W(\mathcal{X},w^{\pm }(\omega ))=\frac{1}{2}\underset{{\rho }_0,{\rho }_1}{max}[1+\parallel \mathcal{X}(H_{\omega }({\rho }_0,{\rho }_1)){\parallel }_1],$$ 3.1

where ∥⋅∥₁ denotes the operator 1-norm.

It is easy to see that without loss of generality one can take ρ₀ and ρ₁ such that [ρ₀,ρ₁]=0. Indeed, let {|k〉} be a basis of eigenvectors of the Helstrom matrix H_ω(ρ₀,ρ₁). The complete dephasing channel ${\mathcal{F}}_d^0$ on the basis {|k〉} is such that

$$H_{\omega }({\rho }_0,{\rho }_1)={\mathcal{F}}_d^0(H_{\omega }({\rho }_0,{\rho }_1))=H_{\omega }({\sigma }_0,{\sigma }_1),$$

where ${\sigma }_i:={\mathcal{F}}_d^0({\rho }_i)$ and therefore [σ₀,σ₁]=0. By applying channel $\mathcal{X}$ we have the following identity

$$\mathcal{X}(H_{\omega }({\rho }_0,{\rho }_1))=\mathcal{X}(H_{\omega }({\sigma }_0,{\sigma }_1)).$$

Therefore, the encoding {σ_i} performs as well as the encoding {ρ_i}, and thus without loss of generality we can take the supremum in equation (3.1) over commuting encodings only.

Moreover, one can see that without loss of generality one can take σ_i to be orthogonal pure states. Indeed, let ${\sigma }_i=\sum _k{\mu }_{k|i}|k⟩⟨k|$ be a spectral decomposition of σ_i. Owing to the convexity of the trace norm, we have

$$\begin{array}{rl}\parallel \mathcal{X}(H_{\omega }({\sigma }_0,{\sigma }_1)){\parallel }_1& ={\parallel \sum _{k,l}{\mu }_{k|0}{\mu }_{l|1}\mathcal{X}(H_{\omega }(|k⟩⟨k|,|l⟩⟨l|))\parallel }_1\\ & \le \sum _{k,l}{\mu }_{k|0}{\mu }_{l|1}\parallel \mathcal{X}(H_{\omega }(|k⟩⟨k|,|l⟩⟨l|)){\parallel }_1\\ & \le \underset{k,l}{max}\parallel \mathcal{X}(H_{\omega }(|k⟩⟨k|,|l⟩⟨l|)){\parallel }_1.\end{array}$$

Then we have the following preliminary result.

Lemma 3.2 —

The maximum in equation (2.2) is given by an orthonormal pure encoding, namely

$$W(\mathcal{X},w^{\pm }(\omega )):=\underset{\begin{array}{c}|{\varphi }_0⟩,|{\varphi }_1⟩\\ ⟨{\varphi }_1|{\varphi }_0⟩=0\end{array}}{max}\frac{1}{2}[1+\parallel \mathcal{X}(H_{\omega }({\varphi }_0,{\varphi }_1)){\parallel }_1]$$

and by an orthogonal POVM such that π₀ is the projector on the positive part of H_ω(ϕ₀,ϕ₁) and ${\pi }_1=𝟙-{\pi }_0$.

Here, for any pure state |ϕ〉 we denote with ϕ:=|ϕ〉〈ϕ| the corresponding projector.

4. Dihedrally covariant qubit channel

Let us start with the case where $\mathcal{X}:\mathcal{L}(\mathcal{H})\to \mathcal{L}(\mathcal{K})$ is a qubit channel, i.e. $\dim \mathcal{H}=\dim \mathcal{K}=2$. Since Pauli matrices span the space of qubit Hermitian operators, any qubit state ρ can be parametrized in terms of Pauli matrices, i.e.

$$\rho =\frac{1}{2}(𝟙+{\mathit{\sigma }}^{\mathrm{T}}\cdot \mathit{x}),|\mathit{x}{|}_2\le 1,$$ 4.1

where σ=(σ_x,σ_y,σ_z)^T and x are the vectors of Pauli matrices and their real coefficients, respectively. Analogously, any qubit channel $\mathcal{X}$ can be parametrized in terms of Pauli matrices, i.e.

$$\mathcal{X}(\rho )=\frac{1}{2}(𝟙+{\mathit{\sigma }}^{\mathrm{T}}\cdot (A\mathit{x}+\mathit{b})),$$

where $A_{i,j}=\frac{1}{2}\mathrm{Tr}[{\sigma }_i\mathcal{X}({\sigma }_j)]$ and $b_i=\frac{1}{2}\mathrm{Tr}[{\sigma }_i\mathcal{X}(𝟙)]$.

With such a parametrization $\mathcal{X}(H_{\omega }({\varphi }_0,{\varphi }_1))$ assumes a very simple form given by

$$\mathcal{X}(H_{\omega }({\varphi }_0,{\varphi }_1))=\frac{1}{2}[\omega 𝟙+{(A\mathit{x}+\omega \mathit{b})}^{\mathrm{T}}\cdot \mathit{\sigma }],$$

whose eigenvalues are $\frac{1}{2}(\omega \pm |A\mathit{x}+\omega \mathit{b}{|}_2)$. Thus, the witness threshold $W(\mathcal{X},w^{\pm }(\omega ))$ in equation (2.2) can be readily computed by means of lemma 3.2 as

$$W(\mathcal{X},w^{\pm }(\omega ))=\frac{1}{2}[1+max(|\omega |,\underset{\begin{array}{c}\mathit{x}\\ |\mathit{x}{|}_2\le 1\end{array}}{max}|A\mathit{x}+\omega \mathit{b}{|}_2)].$$

Note that this expression is the maximum between two strategies. The first one is given by the trivial POVM and thus corresponds to trivial guessing. The second one can be further simplified by means of the following substitutions. Let A=VDU be a polar decomposition of matrix A with U and V unitaries and D diagonal and positive-semidefinite with eigenvalues d (accordingly c:=−V ^†b). By unitary invariance of the 2-norm one has

$$\underset{\begin{array}{c}\mathit{x}\\ |\mathit{x}{|}_2\le 1\end{array}}{max}|A\mathit{x}+\omega \mathit{b}{|}_2=\underset{\begin{array}{c}\mathit{x}\\ |\mathit{x}{|}_2\le 1\end{array}}{max}|D\mathit{x}-\omega \mathit{c}{|}_2.$$

By defining y:=Dx one has

$$\underset{\begin{array}{c}\mathit{x}\\ |\mathit{x}{|}_2\le 1\end{array}}{max}|D\mathit{x}-\omega \mathit{c}{|}_2=\underset{\begin{array}{c}\mathit{y},\mathit{z}\\ |D^{-1}\mathit{y}+(𝟙-D^{-1}D)\mathit{z}{|}_2\le 1\end{array}}{max}|\mathit{y}-\omega \mathit{c}{|}_2,$$

where (⋅)⁻¹ denotes the Moore–Penrose pseudoinverse. By explicit computation it follows that ${[D^{-1}]}^{\mathrm{T}}(𝟙-D^{-1}D)=0$, and therefore vectors D⁻¹y and $(𝟙-D^{-1}D)\mathit{z}$ are orthogonal. Then for any optimal (y,z) one has that (y,0) is also optimal, since $|D^{-1}\mathit{y}+(𝟙-D^{-1}D)\mathit{z}{|}_2\ge |D^{-1}\mathit{y}{|}_2$. Therefore, we have

$$W(\mathcal{X},w^{\pm }(\omega ))=\frac{1}{2}[1+max(\omega ,\mathrm{\Delta }(\omega ))],$$ 4.2

where

$$\mathrm{\Delta }(\omega ):=\underset{\begin{array}{c}\mathit{y}\\ |D^{-1}\mathit{y}{|}_2\le 1\end{array}}{max}|\mathit{y}-\omega \mathit{c}{|}_2.$$ 4.3

The maximum in equation (4.3) is a quadratically constrained quadratic optimization problem, which is known to be NP-hard in general. However, Δ(ω) has a simple geometrical interpretation: it is the maximum Euclidean distance of vector ωc and ellipsoid |D⁻¹y|₂≤1. This interpretation suggests symmetries under which the optimization problem becomes feasible. In particular, we take vector c to be parallel to one of the axes of the ellipsoid |D⁻¹y|₂≤1, namely c₁=c₂=0 (up to irrelevant permutations of the computational basis).

This configuration corresponds to a D₂-covariant channel $\mathcal{X}$, where D₂ is the dihedral group of the symmetries of a line segment, consisting of two reflections and a π-rotation. This configuration is depicted in figure 1. In particular, a qubit channel $\mathcal{X}$ is D₂-covariant if and only if there exist unitary representations $U_k\in {\mathbb{R}}^{3\times 3}$ and $V_k\in {\mathbb{R}}^{3\times 3}$ of D₂ such that

$$A{U}_k\mathit{x}+\mathit{b}=V_k(A\mathit{x}+\mathit{b}).$$ 4.4

Up to unitaries, the most general unitary representation of D₂ in ${\mathbb{R}}^{3\times 3}$ is given by

$$W_1={\sigma }_z\oplus 1,W_2=-{\sigma }_z\oplus 1\mathrm{and}{W}_3=-𝟙\oplus 1,$$

where W₁ and W₂ are reflections and W₃ is a π-rotation. We take U_k:=U^†W_kU and V _k:=V W_kV ^†. Then by explicit computation we have

$$A{U}_k\mathit{x}+\mathit{b}=V_{k}A\mathit{x}+\mathit{b},$$

where we used the fact that [D,W_k]=0 for any k. Therefore, D₂ covariance expressed by equation (4.4) is equivalent to the requirement W_kc=c, namely c₁=c₂=0.

Figure 1.

Bloch-sphere representation of: (a,b) dihedrally covariant channels $\mathcal{X}$ mapping the sphere into an ellipsoid (a) centred in the Bloch sphere (e.g. any Pauli channel ${\mathcal{P}}^{\mathit{\lambda }}$), or (b) translated by a vector c which is parallel to one of the axis of the ellipsoid (e.g. any amplitude damping channel ${\mathcal{A}}^{\mathrm{\lambda }}$); (c) non-dihedrally covariant channel $\mathcal{X}$, as the ellipsoid is translated by a vector c which is not parallel to any of the axis of the ellipsoid. (Online version in colour.)

Under the assumption of D₂-covariance, we take without loss of generality d₂≥d₁ and c₃≥0. If also c₃=0, we further take without loss of generality d₃≥d₂. Then, as formally proved in the appendix, the maximum Euclidean distance Δ(ω) in equation (4.3) can be explicitly computed, leading to the following result.

Lemma 4.1 —

The witness threshold $W(\mathcal{X},w^{\pm }(\omega ))$ of any qubit D₂-covariant channel $\mathcal{X}$ is given by equation (4.2) where

$$\mathrm{\Delta }(\omega )=\{\begin{array}{ll}{d}_2\sqrt{1+\frac{c_3^2{\omega }^2}{d_2^2-d_3^2}},& \text{if }|\omega |<\frac{d_2^2-d_3^2}{d_3{c}_3},\\ d_3+c_3|\omega |,& \mathit{ otherwise.}\end{array}$$

The optimal encoding is given by equation (4.1) with x=D⁻¹y and

$$\mathit{y}=\{\begin{array}{ll}{(0,\pm d_2\sqrt{1-\frac{c_3^2{d}_3^2{\omega }^2}{{(d_3^2-d_2^2)}^2}},\frac{c_3{d}_3^2\omega }{d_3^2-d_2^2})}^{\mathrm{T}}& \text{if }|\omega |\le \frac{d_2^2-d_3^2}{d_3{c}_3}\\ {(0,0,\pm d_3)}^{\mathrm{T}}& \mathit{ otherwise}.\end{array}$$

Using lemmas 4.1 and 2.1, equation (2.3) becomes the maximum over ω of the minimum of two functions. The maximum is attained either in the maxima 0, ±ω₁, or ±1 of the two functions over the domain [−1,1], where

$${\omega }_1:=\frac{(d_2^2-d_3^2)(p_{1|1}-p_{2|2})}{c_3\sqrt{c_3^2{d}_2^2-(d_2^2-d_3^2){(p_{1|1}-p_{2|2})}^2}},$$

(the limit should be considered if c₃=0), or in their intersection ±ω₂ given by

$${\omega }_2:=\{\begin{array}{ll}\sqrt{\frac{d_2^2(d_2^2-d_3^2)}{d_2^2-d_3^2-d_2^2{c}_3^2}},& \text{if }(d_2^2-d_3^2)>d_2^2{c}_3,\\ \frac{d_3}{1-c_3},& \text{ otherwise.}\end{array}$$

We can then state our first main result, formally proved in the appendix, namely a complete and closed-form characterization of the set $S_2^2(\mathcal{X})$ of conditional probability distributions compatible with any qubit D₂-covariant channel $\mathcal{X}$.

Theorem 4.2 —

Any given binary conditional probability distribution p is compatible with any given qubit D₂-covariant channel $\mathcal{X}$ if and only if

$$\underset{\omega \in \mathit{\Omega }}{max}(p^{\mathrm{T}}\cdot w^{\pm }(\omega )-W(\mathcal{X},w^{\pm }(\omega )))\le 0,$$ 4.5

where Ω:={0,±ω₁,±ω₂,±1}∩[−1,1].

As applications of theorem 4.2, let us explicitly characterize the sets of binary conditional probability distributions compatible with two relevant examples of qubit D₂-covariant channels: the Pauli and amplitude-damping channels.

Any Pauli channel can be written as ${\mathcal{P}}^{\mathit{\lambda }}:\rho \to {\mathrm{\lambda }}_0\rho +\sum _{k=1}^3{\mathrm{\lambda }}_k{\sigma }_k\rho {\sigma }_k^{†}$, where σ=(σ_x,σ_y,σ_z) are the Pauli matrices. One has that c₃=0 and $d_3=\underset{k\in [1,3]}{max}|2({\mathrm{\lambda }}_0+{\mathrm{\lambda }}_k)-1|\ge d_2$, thus ${\omega }_1=\mathrm{\infty }$ and ω₂=d₃ and the maximum in equation (4.5) is attained for ω=±ω₂. Thus, upon applying theorem 4.2, one has the following result.

Corollary 4.3 —

Any given binary conditional probability distribution p is compatible with the Pauli channel ${\mathcal{P}}^{\mathit{\lambda }}$ if and only if

$$\frac{|p_{1|1}-p_{1|2}|}{1-|p_{1|1}-p_{2|2}|}\le \underset{k\in [1,3]}{max}|2({\mathrm{\lambda }}_0+{\mathrm{\lambda }}_k)-1|.$$

Any amplitude-damping channel can be written as ${\mathcal{A}}^{\mathrm{\lambda }}(\rho )=\sum _{k=0}^1{A}_k\rho A_k^{†}$, where $A_0=|0⟩⟨0|+\sqrt{\mathrm{\lambda }}|1⟩⟨1|$ and $A_1=\sqrt{1-\mathrm{\lambda }}|0⟩⟨1|$. As shown in the appendix, one has that c₃=1−λ and d₃=λ, $d_2=d_1=\sqrt{\mathrm{\lambda }}$, and thus the maximum in equation (4.5) is attained for ω=±ω₁ or ω=±1. Thus, upon applying theorem 4.2, one has the following result, formally proved in the appendix.

Corollary 4.4 —

Any given binary conditional probability distribution p is compatible with the amplitude-damping channel ${\mathcal{A}}^{\mathrm{\lambda }}$ if and only if

$${(\sqrt{p_{1|2}{p}_{2|1}}-\sqrt{p_{1|1}{p}_{2|2}})}^2\le \mathrm{\lambda }.$$

5. Universally-covariant commutativity-preserving channels

Let us now move to the arbitrary dimensional case. We trade generality regarding the dimension for generality regarding the symmetry of the channel, and assume universal covariance. A channel $\mathcal{X}:\mathcal{L}(\mathcal{H})\to \mathcal{L}(\mathcal{K})$ is universally covariant if and only if there exist unitary representations $U_g\in \mathcal{L}(\mathcal{H})$ and $V_g\in \mathcal{L}(\mathcal{K})$ of the special unitary group SU(d) with $d:=\dim \mathcal{H}$, such that for every state $\rho \in \mathcal{L}(\mathcal{H})$ one has

$$\mathcal{X}(U_g\rho U_g^{†})=V_g\mathcal{X}(\rho )V_g^{†}.$$ 5.1

From universal covariance it immediately follows that any orthonormal pure encoding attains the witness threshold $W(\mathcal{X},w^{\pm }(\omega ))$ in equation (3.1). Indeed, for any orthonormal pure states {ϕ_i} let U be the unitary such that ϕ_i=U|i〉〈i|U^†. Then one has

$$\begin{array}{rl}\parallel \mathcal{X}(H_{\omega }({\varphi }_0,{\varphi }_1)){\parallel }_1& =\parallel \mathcal{X}(H_{\omega }(U|0⟩⟨0|U^{†},U|1⟩⟨1|U^{†})){\parallel }_1\\ & =\parallel V\mathcal{X}(H_{\omega }(|0⟩⟨0|,|1⟩⟨1|))V^{†}{\parallel }_1\\ & =\parallel \mathcal{X}(H_{\omega }(|0⟩⟨0|,|1⟩⟨1|)){\parallel }_1,\end{array}$$

where the second equality follows from equation (5.1), and the third from the invariance of trace distance under unitary transformations. Then we have the following result.

Lemma 5.1 —

The witness threshold $W(\mathcal{X},w^{\pm }(\omega ))$ of any universally covariant channel $\mathcal{X}$ is given by

$$W(\mathcal{X},w^{\pm }(\omega ))=\frac{1}{2}[1+\parallel \mathcal{X}(H_{\omega }(|0⟩⟨0|,|1⟩⟨1|)){\parallel }_1].$$ 5.2

The optimal encoding is given by any pair of orthonormal pure states.

Equation (5.2) has a simple dependence on w in the case when channel $\mathcal{X}$ is commutativity preserving, i.e. $[\mathcal{X}({\rho }_0),\mathcal{X}({\rho }_1)]=0$ whenever $[{\rho }_0,{\rho }_1]=0$. Notice that it suffices to check commutativity preservation for pure states, indeed a channel $\mathcal{X}$ is commutativity preserving if and only if $[\mathcal{X}({\varphi }_0),\mathcal{X}({\varphi }_1)]=0$ whenever 〈ϕ₁|ϕ₀〉=0. Necessity is trivial, and sufficiency follows by assuming $[{\rho }_0,{\rho }_1]=0$, and considering a simultaneous spectral decompositions of ${\rho }_0=\sum _k{\mu }_k{\varphi }_k$ and ${\rho }_1:=\sum _j{\nu }_j{\varphi }_j$. Then one has

$$\begin{array}{rl}[\mathcal{X}({\rho }_0),\mathcal{X}({\rho }_1)]& =\sum _{k,l}{\mu }_k{\nu }_l[\mathcal{X}({\varphi }_k),\mathcal{X}({\varphi }_l)]\\ & =0,\end{array}$$

where the last inequality follows from the fact that 〈ϕ_l|ϕ_k〉=δ_k,l. For a universally covariant channel $\mathcal{X}$, it immediately follows from equation (5.1) that it suffices to check commutativity preservation for an arbitrary pair of orthogonal pure states.

In this case $\mathcal{X}(|0⟩⟨0|)$ and $\mathcal{X}(|1⟩⟨1|)$ admit a common basis of eigenvectors {|k〉}, and thus a spectral decomposition of the Helstrom matrix $\mathcal{X}(H_{\omega }(|0⟩⟨0|,|1⟩⟨1|))$ is given by

$$\mathcal{X}(H_{\omega }(|0⟩⟨0|,|1⟩⟨1|))=\sum _k({\alpha }_k\omega +{\beta }_k)|k⟩⟨k|,$$

where α_k and β_k are the half-sum and half-difference of the kth eigenvectors of $\mathcal{X}(|0⟩⟨0|)$ and $\mathcal{X}(|1⟩⟨1|)$, respectively. Therefore equation (5.2) becomes

$$W(\mathcal{X},w^{\pm }(\omega ))=\frac{1}{2}(1+\sum _k|{\alpha }_k\omega +{\beta }_k|).$$

Then, the optimization problem in equation (2.3) becomes piece-wise linear, thus the maximum is attained on the intersections of the piece-wise components given by γ_k:=β_k/α_k when such values belongs to the domain [−1,1], or on its extrema. We can then provide our second main result, namely a complete closed-form characterization of the set $S_2^2(\mathcal{X})$ of conditional probability distributions compatible with any arbitrary-dimensional universally-covariant commutativity-preserving channel $\mathcal{X}$.

Theorem 5.2 —

Any given binary conditional probability distribution p is compatible with any given arbitrary-dimensional universally-covariant commutativity-preserving channel $\mathcal{X}$ if and only if

$$\begin{array}{rl}|p_{1|1}-p_{1|2}|& \le \sum _k|{\beta }_k|,\\ |p_{1|1}-p_{1|2}|& \le \parallel \mathcal{X}(H_{{\gamma }_k}(|0⟩⟨0|,|1⟩⟨1|)){\parallel }_1-{\gamma }_k|p_{1|1}-p_{2|2}|,\end{array}$$

for any k such that γ_k∈[−1,1].

As applications of theorem 5.2, let us explicitly compute the binary conditional probability distributions compatible with any erasure, depolarizing, universal optimal 1→2 cloning and universal optimal transposition channels. As discussed before, commutativity preservation can be immediately verified for all of these channels by checking that $[\mathcal{X}(|0⟩⟨0|),\mathcal{X}(|1⟩⟨1|)]=0$.

Any erasure channel can be written as ${\mathcal{E}}_d^{\mathrm{\lambda }}:\rho \to \mathrm{\lambda }\rho \oplus (1-\mathrm{\lambda })\varphi$, where ϕ is some pure state. One can compute that α=(λ/2,λ/2,0×d−2,1−λ) and β=(λ/2,−λ/2,0×d−1), thus upon applying theorem 5.2 one has the following corollary.

Corollary 5.3 —

Any given binary conditional probability distribution p is compatible with the erasure channel ${\mathcal{E}}_d^{\mathrm{\lambda }}$ if and only if

$$|p_{1|1}-p_{1|2}|\le \mathrm{\lambda }.$$

Any depolarizing channel can be written as ${\mathcal{D}}_d^{\mathrm{\lambda }}:\rho \to \mathrm{\lambda }\rho +(1-\mathrm{\lambda })(𝟙/d)$. One can compute that α=(λ/2+(1−λ)/d×2,(1−λ)/d×d−2) and β=(−λ/2,λ/2,0×d−2), thus upon applying theorem 5.2 one has the following corollary.

Corollary 5.4 —

Any given binary conditional probability distribution p is compatible with the depolarizing channel ${\mathcal{D}}_d^{\mathrm{\lambda }}$ if and only if

$$|p_{1|1}-p_{1|2}|\le \mathrm{\lambda }$$

and

$$\frac{|p_{1|1}-p_{1|2}|}{1-|p_{1|1}-p_{2|2}|}\le \frac{d\mathrm{\lambda }}{2-2\mathrm{\lambda }+d\mathrm{\lambda }}.$$

The universal optimal 1→2 cloning channel can be written as ${\mathcal{C}}_d^{\mathrm{\lambda }}:\rho \to (2/(d+1))P_S(\rho \otimes 𝟙)P_S$. By explicit computation one has

$${\mathcal{C}}_d(|i⟩⟨i|)=\frac{1}{2(d+1)}\sum _k(|k,i⟩+|i,k⟩)(⟨k,i|+⟨i,k|),$$

and therefore $[{\mathcal{C}}_d(|0⟩⟨0|),{\mathcal{C}}_d(|1⟩⟨1|)]=0$, thus the universal optimal 1→2 cloning ${\mathcal{C}}_d$ is a commutativity preserving channel. One can compute that α=(1/(d+1)×3,1/2(d+1)×2(d−2)) and β=(−1/(d+1),1/(d+1),0,−1/2(d+1)×d−2,1/2(d+1)×d−2), thus upon applying theorem 5.2 one has the following corollary.

Corollary 5.5 —

Any given binary conditional probability distribution p is compatible with the universal optimal 1→2 cloning channel ${\mathcal{C}}_d$ if and only if

$$|p_{1|1}-p_{1|2}|\le \frac{d}{d+1}.$$

The universal transposition channel can be written as ${\mathcal{T}}_d:\rho \to (1/(d+1))({\rho }^{\mathrm{T}}+𝟙)$. One can compute that α=(3/2(d+1)×2,1/(d+1)×d−2) and β=(1/2(d+1),−1/2(d+1),0×d−2), thus upon applying theorem 5.2 one has the following corollary.

Corollary 5.6 —

Any given binary conditional probability distribution p is compatible with the universal transposition channel ${\mathcal{T}}_d$ if and only if

$$|p_{1|1}-p_{1|2}|\le \frac{1}{d+1}$$

and

$$\frac{|p_{1|1}-p_{1|2}|}{1-|p_{1|1}-p_{2|2}|}\le \frac{1}{3}.$$

The results of corollaries 4.3, 4.4, 5.3–5.6 are summarized in table 1.

Table 1.

Complete closed-form characterization of the set $S_2^2(\mathcal{X})$ of binary conditional probability distributions compatible with channel $\mathcal{X}$, for $\mathcal{X}$ given by the Pauli channel ${\mathcal{P}}^{\mathit{\lambda }}$, the amplitude damping channel ${\mathcal{A}}^{\mathrm{\lambda }}$, the erasure channel ${\mathcal{E}}_d^{\mathrm{\lambda }}$, the depolarizing channel ${\mathcal{D}}_d^{\mathrm{\lambda }}$, the universal 1→2 cloning channel ${\mathcal{C}}_d$ and the universal transposer ${\mathcal{T}}_d$, as given by corollaries 4.3, 4.4, 5.3–5.6, respectively.

6. Cartesian representation

In this section, we provide a geometrical interpretation of our results. Binary conditional probability distributions are represented by 2×2 real matrices, so they can be regarded as vectors in ${\mathbb{R}}^4$. However, due to the normalization constraint $\sum _j{p}_{j|i}=1$ for any i, they all lie in a bidimensional affine subspace. A natural Cartesian parametrization of such a subspace is given by

$$p_{j|i}=p(x,y)=\frac{1}{2}[\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)+x\left(\begin{array}{cc}1& -1\\ 1& -1\end{array}\right)+y\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)]$$ 6.1

and binary conditional probability distributions form the square |x±y|≤1, whose four vertices are the right-stochastic matrices with all entries equal to 0 or 1.

As it is clear from equation (6.1):

• — a permutation of the states {ρ₀,ρ₁} corresponds to the transformation (x,y)→(x,−y);

• — a permutation of the effects {π₀,π₁} corresponds to the transformation (x,y)→(−x,−y);

• — a permutation of the states {ρ₀,ρ₁} and effects {π₀,π₁} corresponds to the transformation (x,y)→(−x,y).

Therefore, for any channel $\mathcal{X}$, the set $S_2^2(\mathcal{X})$ of binary conditional probability distributions compatible with $\mathcal{X}$ is symmetric for reflections around the x or y axes (i.e. it is D₂-covariant).

As a consequence of our previous results, the sets $S_2^2({\mathcal{U}}_d)$ and $S_2^2({\mathcal{F}}_d^{\mathrm{\lambda }})$ of conditional probability distributions compatible with any unitary and dephasing channels ${\mathcal{U}}_d$ and ${\mathcal{F}}_d^{\mathrm{\lambda }}$ coincide with the square |x±y|≤1, for any d and any λ. The set $S_2^2(\mathcal{T})$ of conditional probability distributions compatible with any trace-class channel $\mathcal{T}$ coincide with the segment x∈[−1,1], y=0.

With the parametrization in equation (6.1), the sets of binary conditional probability distributions compatible with any Pauli, amplitude-damping, erasure, depolarizing, universal 1→2 cloning and universal transposition channels as given by corollaries 4.3, 4.4, 5.3–5.6, respectively, are given in table 2 and depicted in figure 2.

Table 2.

Cartesian parametrization of the set $S_2^2(\mathcal{X})$ of binary conditional probability distributions compatible with channel $\mathcal{X}$, for $\mathcal{X}$ given by the Pauli channel ${\mathcal{P}}^{\mathit{\lambda }}$, the amplitude damping channel ${\mathcal{A}}^{\mathrm{\lambda }}$, the erasure channel ${\mathcal{E}}_d^{\mathrm{\lambda }}$, the depolarizing channel ${\mathcal{D}}_d^{\mathrm{\lambda }}$, the universal 1→2 cloning channel ${\mathcal{C}}_d$ and the universal transposer ${\mathcal{T}}_d$.

Figure 2.

Cartesian representation of the space of binary conditional probability distributions p. The outer white square denotes the polytope of all binary conditional probability distributions. The inner yellow region denotes the sets $S_2^2(\mathcal{X})$ of conditional probability distributions compatible with: (a) the erasure channel $\mathcal{X}={\mathcal{E}}_d^{\mathrm{\lambda }}$ (for Δy=λ) and the universal optimal 1→2 cloning channel $\mathcal{X}={\mathcal{C}}_d$ (for Δy=d/(d+1)); (b) the Pauli channel $\mathcal{X}={\mathcal{P}}^{\mathit{\lambda }}$ (for $\mathrm{\Delta }y=\underset{k\in [1,3]}{max}|2({\mathrm{\lambda }}_0+{\mathrm{\lambda }}_k)-1|$); (c) the depolarizing channel $\mathcal{X}={\mathcal{D}}_d^{\mathrm{\lambda }}$ (for Δx=((d−2)/d)(1−λ) and Δy=λ) and the universal optimal transposition channel ${\mathcal{T}}_d$ (for Δx=(d−2)/(d+1) and Δy=1/(d+1)); (d) the amplitude-damping channel ${\mathcal{A}}^{\mathrm{\lambda }}$ (for Δy₁=λ and $\mathrm{\Delta }{y}_2=\sqrt{\mathrm{\lambda }}$). (Online version in colour.)

7. Conclusion and outlook

In this work, we developed a device-independent framework for testing quantum channels. The problem was framed as a game involving an experimenter, claiming to be able to produce some quantum channel, and a theoretician, willing to trust observed correlations only. The optimal strategy consists of (i) all the input states and measurements generating the extremal correlations that the experimenter needs to produce and (ii) a full closed-form characterization of the correlations compatible with the claim, that the theoretician needs to compare with the observed correlations. For binary correlations, we explicitly derived the optimal strategy for the cases where the claimed channel is a dihedrally-covariant qubit channel, such as any Pauli and amplitude-damping channels, or an arbitrary-dimensional universally-covariant commutativity-preserving channel, such as any erasure, depolarizing, universal cloning and universal transposition channels.

Natural generalization of our results include relaxing the restriction of binary correlations, that is m=n=2, and extending the characterization of $S_m^n(\mathcal{X})$ to other classes of channels. An interesting generalization would consist of letting the POVM {π_y} depend upon an input not known during the preparation of {ρ_x}, as is the case in quantum random access codes. Moreover, the set-up in equation (1.1) could be modified to allow for entanglement alongside $\mathcal{X}$, or many parallel or sequential uses of channel $\mathcal{X}$.

We conclude by remarking that our results are particularly suitable for experimental implementation. For any channel $\mathcal{X}$ an experimenter claims to be able to produce, our framework only requires them to prepare orthogonal pure input states and perform orthogonal measurements in order to fully characterize $S_2^2(\mathcal{X})$ and thus device-independently test $\mathcal{X}$.

Appendix A. Proofs

In this section, we prove those results reported in the previous sections for which the proof, being lengthy and not particularly insightful, has only been outlined. The numbering of statements follows that of the previous sections.

Lemma A.1. —

The witness threshold $W(\mathcal{X},w^{\pm }(\omega ))$ of any qubit D₂-covariant channel $\mathcal{X}$ is given by equation (4.2) where

$$\mathrm{\Delta }(\omega )=\{\begin{array}{ll}{d}_2\sqrt{1+\frac{c_3^2{\omega }^2}{d_2^2-d_3^2}},& \text{ if }|\omega |<\frac{d_2^2-d_3^2}{d_3{c}_3},\\ d_3+c_3|\omega |,& \mathit{otherwise.}\end{array}$$

Proof. —

Under the assumption of D₂-covariance, take without loss of generality c=(0,0,c₃)^T. Then without loss of generality we take d₂≥d₁ and c₃≥0. If c₃=0 without loss of generality we also take d₃≥d₂.

First notice that y*, which attains the maximum in equation (4.2), lies in the yz plane. Indeed, any ellipse obtained as the intersection of the ellipsoid |D⁻¹y|₂≤1 and a plane containing the z-axis is, up to a z rotation, a subset of the ellipse obtained as the intersection of the ellipsoid |D⁻¹y|₂≤1 and the yz plane.

The generic vector on the boundary of the yz ellipse can be parametrized as

$$\mathit{y}={(0,\pm d_2\sqrt{1-\frac{z^2}{d_3^2}},z)}^{\mathrm{T}},$$

with z∈[−d₃,d₃] and thus the maximum Euclidean distance in equation (4.3) is given by

$$\mathrm{\Delta }(\omega )=\underset{z\in [-d_3,d_3]}{max}\sqrt{d_2^2(1-\frac{z^2}{d_3^2})+{(z-\omega c_3)}^2}.$$ A 1

By explicit computation one has

$$\frac{\mathrm{d}\mathrm{\Delta }(\omega )}{\mathrm{d}z}={[d_2^2(1-\frac{z^2}{d_3^2})+{(z-\omega c_3)}^2]}^{-1/2}[(1-\frac{d_2^2}{d_3^2})z-c_3\omega ],$$

which is zero for $z^{\ast }=c_3{d}_3^2\omega /(d_3^2-d_2^2)$ and

$${\frac{{\mathrm{d}}^2\mathrm{\Delta }(\omega )}{\mathrm{d}{z}^2}|}_{z=z^{\ast }}={\left[d_3^2\sqrt{d_2^2(1+\frac{c_3^2{\omega }^2}{d_2^2-d_3^2})}\right]}^{-1}(d_3^2-d_2^2),$$

namely z* attains the maximum in equation (A.1) whenever d₂≥d₃. Therefore, the maximum is attained by z=z* iff −d₃<z*≤d₃, namely when $|\omega |<(d_2^2-d_3^2)/d_3{c}_3$, and by z=±d₃ otherwise. By replacing z* and ±d₃ in equation (A.1) the statement follows. ▪

Theorem A.2. —

Any given binary conditional probability distribution p is compatible with any given qubit D₂-covariant channel $\mathcal{X}$ if and only if

$$\underset{\omega \in \mathit{\Omega }}{max}(p\cdot w^{\pm }(\omega )-W(\mathcal{X},w^{\pm }(\omega )))\le 0,$$

where Ω:={0,±ω₁,±ω₂,±1}∩[−1,1].

Proof. —

The function $f^{\pm }(\omega ):=p^{\mathrm{T}}\cdot w^{\pm }(\omega )-W(\mathcal{X},\omega )$ is the minimum of continuous functions $g^{\pm }(\omega ):=p^{\mathrm{T}}\cdot w^{\pm }(\omega )-\frac{1}{2}(1+|\omega |)$ and $h^{\pm }(\omega ):=p^{\mathrm{T}}\cdot w^{\pm }(\omega )-\frac{1}{2}(1+\mathrm{\Delta }(\omega ))$. Therefore, $\underset{\omega \in [-1,1]}{max}{f}^{\pm }(x)$ is attained by those values of ω maximizing g^±(ω) or h^±(ω), or in the intersections of g^±(ω) and h^±(ω).

The function g^±(ω) is piece-wise linear and attains its maximum on [−1,1] in 0. The function h^±(ω) is quasi-concave continuous with a continuous derivative. Indeed

$$2\frac{\mathrm{d}{h}^{\pm }(\omega )}{\mathrm{d}\omega }=\{\begin{array}{ll}\pm (p_{1|1}-p_{2|2})-\frac{d_2{c}_3^2\omega }{\sqrt{(d_2^2-d_3^2)(d_2^2-d_3^2+c_3^2{\omega }^2)}},& \text{if }|\omega |<\frac{d_2^2-d_3^2}{d_3{c}_3},\\ \pm (p_{1|1}-p_{2|2})-\mathrm{sgn}(\omega )c_3,& \text{otherwise}\end{array}$$

is continuous and

$$2\frac{{\mathrm{d}}^2{h}^{\pm }(\omega )}{\mathrm{d}{\omega }^2}=\{\begin{array}{ll}-\frac{d_2{c}_3^2{(d_2^2-d_3^2)}^2}{{[(d_2^2-d_3^2)(d_2^2-d_3^2+c_3^2{\omega }^2)]}^{3/2}},& \text{if }|\omega |<\frac{d_2^2-d_3^2}{d_3{c}_3},\\ 0,& \text{if }|\omega |>\frac{d_2^2-d_3^2}{d_3{c}_3}\end{array}$$

is non-positive. Therefore, h^±(ω) attains its maximum on [−1,1] in 0, ±1, or in the zero ±ω₁ of its first derivative.

Owing to the piece-wise linearity of g^±(ω) and the quasi-concavity of h^±(ω), since g^±(0)≥h^±(0) and g^±(±1)≤h^±(±1) one has that g^±(ω) and h^±(ω) intersect in exactly two points ±ω₂∈[−1,1], thus the statement follows. ▪

Corollary A.3. —

Any given binary conditional probability distribution p is compatible with the amplitude-damping channel ${\mathcal{A}}^{\mathrm{\lambda }}$ if and only if

$${(\sqrt{p_{1|2}{p}_{2|1}}-\sqrt{p_{1|1}{p}_{2|2}})}^2\le \mathrm{\lambda }.$$

Proof. —

One has c₃=1−λ, $d_2=\sqrt{\mathrm{\lambda }}$ and d₃=λ, thus

$${\omega }_1=\sqrt{\frac{\mathrm{\lambda }}{(1-\mathrm{\lambda })((1-\mathrm{\lambda })-{(p_{1|1}-p_{2|2})}^2)}}(p_{1|1}-p_{2|2})$$

and ω₂=1. By explicit computation, the conditions ${\omega }_1\in \mathbb{R}$ and |ω₁|≤1 are equivalent to (p_1|1−p_2|2)²<1−λ and (p_1|1−p_2|2)²≤(1−λ)², respectively, thus ω₁∈[−1,1] is equivalent to (p_1|1−p_2|2)²≤(1−λ)² for any λ>0.

By explicit computation, the maximum in equation (4.5) is attained at ω=±ω₁ and ω=±1 whenever |p_1|1−p_2|2|≤1−λ and |p_1|1−p_2|2|>1−λ, respectively. Thus equation (4.5) becomes

$$|p_{1|1}-p_{1|2}|-\sqrt{\mathrm{\lambda }[1-\frac{{(p_{1|1}-p_{2|2})}^2}{1-\mathrm{\lambda }}]}\le 0,$$

whenever |p_1|1−p_2|2|≤1−λ, which, by solving in λ, becomes λ₋≤λ≤λ₊ whenever λ≤1−|p_1|1−p_2|2|, where ${\mathrm{\lambda }}_{\pm }={(\sqrt{p_{1|1}{p}_{2|2}}\pm \sqrt{p_{1|2}{p}_{2|1}})}^2$. By explicit computation 1−|p_1|1−p_2|2|≤λ₊, so the statement follows. ▪

Authors' contributions

All authors contributed equally to the original ideas, analytical derivations and final writing of this manuscript, and gave final approval for publication.

Competing interests

The authors declare no competing interests.

Funding

M.D.A. acknowledges support from the Singapore Ministry of Education Academic Research Fund Tier 3 (grant no. MOE2012-T3-1-009). F.B. acknowledges support from the JSPS KAKENHI, no. 26247016.